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IC 88n 



Bureau of Mines Information Circular/1980 



Methodology for Evaluating 
Integral Gaussian Profiles 

By Arthur B. Campbell III 



»i 






'&* * 









UNITED STATES DEPARTMENT OF THE INTERIOR 






' 



Information Circular 8811 

Methodology for Evaluating 
Integral Gaussian Profiles 

By Arthur B. Campbell III 




UNITED STATES DEPARTMENT OF THE INTERIOR 
Cecil D. Andrus, Secretary 

BUREAU OF MINES 

Lindsay D. Norman, Acting Director 













This publication has been cataloged as follows: 



Campbell, Arthur Benedict III, 1943- 

Methodology for evaluating integral Gaussian profiles. 

(Information circular - U.S. Bureau of Mines ; 8811) 

Bibliography: p. 14-15- 

1. Solids, Effect of radiation on— Data processing. 2. Ion bombard- 
ment—Data processing. 3- X-ray spectroscopy— Data processing. 4. Sput- 
tering (Physics)— Data processing. 1. Title. II. Title: Integral Gaussian 
profiles. III. Series: United States. Bureau of Mines. Information circu- 
lar ; 8811. 



TN295.U4 [QC176.8.R3] 622'.08s [669'. 94] 79-19923 



CONTENTS 

Page 

Abstract 

Introduction 

Experimental techniques 2 

Use of ABC10 program 5 

Example « • 

Conclusions *-3 

References -^ 

Appendix. --Listing of ABC10 16 

ILLUSTRATIONS 

1. Integral data for nickel implanted into iron at 25 kev expressed as 

Ni-L X-ray yield versus sputtering charge 9 

2. Profiles obtained for nickel implanted into iron at 25 kev as 

obtained us ing ABC10 12 

TABLES 

1. Symbols used in ABC10 6 

2. Output of computer and input of data for evaluation of integral 

profiles for nickel implanted into iron at 25 kev and for 

obtaining profiles for as-implanted case 10 

3. Output of computer and input of data for evaluation of integral 

profiles for nickel implanted into iron at 25 kev and for 

obtaining profiles for annealed case 11 

4. Values of range and range straggle for 30-kev As implanted into 

silicon as determined from data of Ludvik, LSS theory, Iwaki , 

and the PIXE-IS technique using ABC10 13 



METHODOLOGY FOR EVALUATING INTEGRAL GAUSSIAN PROFILES 

by 
Arthur B. Campbell 111 ] 



ABSTRACT 

This Bureau of Mines report describes a method that utilizes a computer 
program to evaluate integral Gaussian profiles of atomic concentration versus 
depth in a solid. The specific use illustrated involves profiling ion- 
implanted metals using proton-induced X-ray emission (PIXE) analysis and ion 
sputtering (IS) for sectioning. 

INTRODUCTION 

As part of the Bureau of Mines effort to minimize the requirements for 
minerals through conservation and substitution, an investigation of ion- 
implanted alloys was initiated in 1975. These alloys are an integral part of 
the bulk material, down to depths of about 1000 A, and as such do not have the 
interfacial bonding disadvantages inherent at the junction between substrates 
and coatings. These ion-implanted alloys have been shown to be as corrosion 
resistant as comparable bulk alloys in certain applications (4-5_) , while 
using orders of magnitude less alloying material and thus realizing a consider- 
able conservation. This is especially important since common alloys include 
such strategic elements as chromium and nickel. As a first step in understand- 
ing how ion-implanted alloys resist corrosion, a study was initiated into the 
physical properties of ion-implanted metals,, The distribution of elements 
below the surface was included in this study, which used proton-induced X-ray 
emission (PIXE) analysis combined with sectioning via ion sputtering (IS) o 

The PIXE-IS depth-profiling technique provides data that are in an inte- 
gral form; that is, they represent elemental concentration (areal) below a 
certain depth versus depth «, To convert these data to the more useful form of 
a differential profile of concentration (volume) versus depth, a computer 
program has been developed that produces this depth profile from the integral 
data., 



1 Research physicist, Avondale Research Center, Bureau of Mines, Avondale, Md< 
Underlined numbers in parentheses refer to items in the list of references 
preceding the appendix. 



When actually in use or for intentional redistribution of elements, ion- 
implanted alloys may be subjected to thermal treatments that will alter these 
profiles, so in addition to profiles of as-implanted alloys, it is also neces- 
sary to obtain profiles of alloys that have been annealed. This same computer 
program includes this capability,, 

This Information Circular describes a computerized method for evaluating 
integral Gaussian profiles., It includes enough of the basic theory of ion 
implantation and diffusion that is behind the program so that the usefulness 
of this program to other researchers can be seen,, 

EXPERIMENTAL TECHNIQUES 

A brief summary of the theories of ion implantation profiles and diffu- 
sion and the PIXE-IS technique will better explain the use of this program. 

Ion implantation involves the bombardment of a solid with accelerated 
ions having energies typically in the tens of kiloelectronvolts . These ions 
lose energy in traversing the solid through interactions with the electrons 
and nuclei of the solid, and their final stopping point is determined statis- 
tically,, The most successful theory of ion implantation is that of Lindhard, 
Scharff, and Schiott (LSS) (12) , which has undergone many updatings and modifi- 
cations (16) but is still the same basic statistical system,, To a good 
approximation (±20 percent) , the profile of ion-implanted atoms in a solid 
such as a semiconductor or a metal can be described as 

y(x) =y exp [-<; ( ^/ ], (1) 

where y (x) is the concentration in atoms/cm 3 of the implanted element at a 
depth x, y is the maximum concentration, Rp is the projected range or most 
probable depth, and ARp is the range straggle or standard deviation in the 
range. This equation is valid for implantation at normal incidence to the 
surface of the solid where R p , the range projected along the beam direction, 
will be measured perpendicular to the surface. The LSS theory is strictly 
applicable only in the case of an amorphous solid, that is, in the absence of 
inhomogeneities such as grain boundaries . 

To predict the effect of heat treatment or annealing on the profile as 
given by equation 1, Tick's law of diffusion (6_) is used. Mathematically it 
is expressed as 

M^ti = D a^ti s (2) 

where y(x,t) is the concentration as a function of depth after annealing for 
a time t, where the diffusion coefficient is D in cm /sec. 



A solution to equation 2, assuming a semi-infinite solid with no losses at the 
surface, is (3) 



y(x,t) = -p f y(x + 2VDt) exp (-Tf )dT] 



•n . 



** (3) 



00 

+ -= j y(-x + 2VDt) exp (-Tf)dT) , 



2/Dt 

where y(x,o) = y(x) and T) is simply a variable of integration. Integrating 
equation 3, one obtains 

y(x,t) = (y /2) yi -^(y 2 + y 3 ) , (4) 

where yi = 1 + ^ (5) 

f -(x - Rp) 2 1 _ r-x(ARp) 3 - 2R P Dt "1 ,,. 

^ " exp L 2(ARp)«yx J erfc L 2(AR p ) 2 /Dt/ylJ (6) 

v = exp |" -(x+ Rp) 3 1 f r x(AR P ) 2 - 2R P Dt l () 

y3 exp L 2(AR p ) 2 yiJ erfC L 2(ARp)2 v ^/7I J C ; 

where erfc is the complementary error function. Note that equation 4 is 
finite as both x and t approach infinity. It can be seen that equation 4 
reduces to equation 1 for t = o. Since we have assumed an impenetrable bar- 
rier at the surface, y can be defined from the definition of 0, the implanted 
f luence (or dose) , in atoms/cm . 

CO 

= J y(x,o)dx. (8) 

o 
With a change of variable and including equation 4, this becomes 

oo -1 

Ye = ^— [ exp (-z) 2 dzl = —M \l + erf( ** : Y l , (9) 

y /2(ARp) U_ Rp J /2^(AR P )L V/2(AR P ) VJ' 



/2(ARp) 
where erf is the error function. 
Putting this into equation 4 



'<*>« ' ZfW l l * "<-7m£ti y '" % (y = + ^ • (10) 



which is the complete solution. 



The PIXE-IS technique involves the use of ion beams for both analysis and 
sectioning. Beams of high-energy (100- 200-kev) protons bombarding the sur- 
face produce X-rays characteristic of the elements present in a thin surface 
layer (=-1000 to 5000 A). These X-rays can also be used to obtain quantita- 
tive information (14) . Thus, it is possible to obtain a conversion from X-ray 
yield (X-ray photons measured per microcoulomb of proton beam) to areal concen- 
tration (atoms of element per square centimeter of surface). For thin enough 
films (11) and with a knowledge of the density of the film, atomic concentra- 
tions (in atoms per cubic centimeter) can be obtained. 

Low-energy (=-1 kev) inert gas ions can interact with a surface such that 
atoms of the surface are ejected. This process of physical sputtering is used 
to controllably remove thin layers from the surface, even as thin as one 
atomic layer. This process can thus be used to depth-profile a surface. 

The profiles obtained by the PIXE-IS technique are of an integral form; 
that is, they are a measure of the integral areal concentration or amount left 
in the surface versus depth. The process involves measuring via PIXE the 
X-ray photons per microcoulomb of a particular element, sputtering for a par- 
ticular charge (microcoulombs) of inert gas ions, and then remeasuring the 
X-ray yield. The difference in the X-ray yields is a measure of the amount of 
the element removed by sputtering, assuming the PIXE analysis depth is greater 
than the profile depth „ There will be a limitation from this factor on pro- 
files that are deeper than about 5000 A, which could be important for the case 
of long anneal times. With appropriate conversion factors, the data obtained 
(X-ray yield versus sputtering charge) can be converted to areal concentration 
versus depth. 

The conversion from X-ray yield to areal concentration is accomplished 
using the thin-film approximation (11) , 

T = I/ax , (ID 

where T is the areal concentration in atoms/cm s , I is the X-ray yield, and <j x 
is the X-ray production cross section. The conversion from sputtering charge 
is accomplished using the equation 

d = QCSf a /[Ap(l + Y)] , (12) 

where d is the depth normal to the surface, Q is the accumulated sputtering 
charge, C is a unit conversion factor, S is the sputtering yield in atoms sput- 
tered per ion, f a is a geometrical factor, A p is the area of the proton analy- 
sis beam, and y is the secondary electron emission coefficient during sputtering, 
These factors are measured during preliminary PIXE and IS experiments. 

To convert this integral data to a differential profile of atomic concen- 
tration versus depth, a method had to be developed that would produce a physi- 
cally realistic profile from these data points. Since theoretical predictions 
are for a Gaussian profile shape (equation 1), it was decided to use the 
integral of equation 1 and compare this smooth curve with the data using the 
values of Rp and ARp as parameters for obtaining the best agreement. 



In converting the data from annealed samples, it was decided to use the 
predictions of Fick's law for diffusion of an initially Gaussian profile. 
Since equation 4 reduces to equation 1 for t = 0, the integral of equation 4 

N(d,t) = y(x,t)dx , (13) 

d 

where N(d,t) represents the concentration of an element below a depth d, can 
be used for both as-implanted and annealed samples. 

The integral of equation 4 is accomplished using an approximation for the 
complementary error function (1_) and a numerical integration using the method 
of Gaussian quadrature (7) . 

The complementary error function can be expressed as 

erfc(z) = 1 - erf(z) , (14) 

and the approximation 



e: 



rf(z) =» 1 - [a 1 b + a s b 2 + a 3 b3] exp(-z 2 ) (15) 



with b = 1/(1 + (0.47047)z), a 1 = 0.3480242, a 3 = -0.0958798, and 
a 3 = 0.7478556 (J.) is used for < z <• oo with an accuracy of 



ei 



rf(z) - {1 - [a x b + a 2 b 3 + a 3 b 3 ] exp(-z 2 )} < 2.5x10 (16) 



to evaluate equation 13. 

The General Services Administration's RAMUS time-sharing computer service 
has been used for these evaluations , and it provides a program for numerical 
integration using the Gaussian quadrature method called NUMINT*** (8), which 
was modified for use and is included in this program, as shown in the appendix. 

USE OF ABC 10 PROGRAM 

A runthrough of the program, which was entitled ABC10, will serve to 
explain its workings and aid in understanding how to use it. All line numbers 
refer to those in the ABC10 program printed in the appendix. Note that the 
computer language BASIC is used. In this section we will refer to the symbols 
as they would appear in the printout during actual use of the program. 

In the use of the program, one must first have a graph of the integral 
profile data points. Initializing parameters are first entered into the pro- 
gram, and then intelligent estimates are made as to the values of range (R) 
and range straggle (RS) , and diffusion coefficient (D) in the case of an 
annealed profile. The resulting profile values are compared with the data 
until the best agreement is reached. This method is fitting by inspection. 
Once the best agreement by this method has been reached, one would stop and 
draw a detailed plot of the computer-generated curve on the same graph that 



contains the data points. If the two look similar, a detailed manual analysis 
must be performed to obtain an estimate of the standard deviation. In all 
cases, this method was able to produce integral profiles that agreed with the 
data to no worse than 10 percent standard deviation. 

The initial steps of the program provide for the input of constants and 
data to be used in the evaluations. Table 1 lists the symbols typed out 
during operation, but note that these do not necessarily conform to the nota- 
tion used in the program because of expediencies in programing. The symbols 
are defined later in the text. Statement 120 is a comment that appears first 
in the operation of the program and provides a way of resetting the program 
during use by using certain nonrealistic values of range and range straggle. 
In the normal operation of the program, various values of R, R.S., and D are 
chosen; the resultant integral function is compared with the data, and the 
values that produce the best fit are chosen. In some cases, it is necessary 
to reset the initial parameters , and statement 120 provides the user with a 
reference to the resulting procedure. 



TABLE 1. - Symbols used in ABC10 



Definition (text) 



Symbol in print 



Symbol in program 



Range (R^ ) 

Range straggle (^ ) 

Diffusion coefficient (D) 

Diffusion time (t) , 

Conversion factors: 

Angstroms/10 3 microcoulomb. . . , 
Atoms/cm /photon/microcoulomb. 

Incremental depth 

Infinity (maximum depth v 1.25). 

X-ray absorption factor ( tL p) .. , 



Stopping power 

Power factor , 

Initial proton energy. 



R 

R.S. 
D 
T 

A/K 

A/C/P/MIC . 
STEP SIZE 
MAXIMUM DEPTH 

X-RAY ABSORPTION 
FACTOR (ANG.-l) 
P 
F 
EO 



B 
C 
D 
T 

K 
H 
T6 

V 

Q8 

Gl 
G2 
G3 



The initial parameters entered are the step size and maximum depth: that 
is, the incremental step between integral data points desired and the largest 
depth for an integral data point. This maximum depth is the depth where the 
integral data points reach an asymptotic value. This is not necessarily zero 
because of the nature of the sputtering technique (so called knock-on effects 
or differential sputtering). In a profiling experiment, the sputtering will 
always be continued until at least this asymptotic value is reached. This 
maximum depth is also used to define the upper limit of integration, as dis- 
cussed later. The step size and maximum depth are usually chosen so that the 
integral is evaluated for points on the abscissa that agree with the data 
points . 

The next parameters that are requested are the values of the integral 
data at zero sputtering and after completion of the sputtering. These are 
expressed as the value of the integral at zero and at infinity, and it may be 



necessary to use these two as fitting parameters. For several reasons, 
including peak background and the physics of the sputtering process (15), the 
data may not approach zero with continued sputtering but rather approach an 
equilibrium value. This value would be used as the value at infinity, and the 
program takes this into account in the fitting. 

Also included in this program is a factor that takes into account the 
absorption of X-rays in the solid. If X-rays are produced at some depth x 1 , 
then the intensity of the X-rays leaving the surface is 



I exp 



[-(!>']• (17) 



where I is the intensity of the X-rays as produced, u is the X-ray absorption 
coefficient, and p is the atomic density of the solid. The program evaluates 
the effect of this absorption at each depth interval, and the resultant inte- 
gral profile reflects the profile as it would appear as data; that is, with 
absorption included. Values of the X-ray absorption coefficient are available 

li 
in tabular form (9) , expressed as — . Thus , the program requests the absorp- 

P . 
tion factor, which is the product of ( — j and p, expressed as inverse ang- 
stroms. This somewhat unusual unit is for ease of programing, since the usual 
unit is cm 2 /gram x gram/cm 3 , or inverse centimeters. If negligible absorption 
is expected or to eliminate this correction factor, a value of zero is entered. 

The program also takes into account the changes in X-ray production 
yields because of the energy loss of the protons in traversing the solid. The 
stopping power (P) , the energy loss of the protons per distance traveled, can 
be assumed to be a constant in the energy region considered (2^) , and so the 
instantaneous energy of the protons is 

E(x) = EO - Px' , (18) 

where E is the energy of the protons at a depth x 1 . The best fit to the 
energy dependence of the X-ray production cross section in the energy region 
of interest here was found to be a power function. Then the ratio of cross 
sections at a depth x' and at the surface will provide the correction 
factor (G) , 



= [ ^fo^ ]' <») 



where F is the predetermined power factor. 

In both the case of absorption and the case of energy decrease, the value 
of x 1 chosen for evaluation is one-half the distance from d to infinity. This 
gives an average correction factor, but since these correction factors are not 
a strong function of depth, this is considered adequate. 



The next step in the program is to enter the loop that provides data 
points, which are the integral of a Gaussian profile function based on the 
values of range, range straggle, diffusion coefficient, and diffusion time. 
If the profile is for an unannealed sample, t is set to zero. Note that a 
nonzero value of D must be entered even though it is not used. 

In normal use, the procedure that is followed from here on for unannealed 
samples involves choosing various values of R and R.S. and comparing the 
resultant values of the integral Gaussian with the data points in order to get 
the best fit. This procedure is best accomplished with a graph of the data 
in front of the computer operator. For a detailed comparison, the operator 
must go off line and make careful plots of the integral Gaussian functions 
on the same graph. Manual statistical analysis at this point is time consum- 
ing, but for the one that is the "best" fit it will confirm the visual 
comparison. 

Nothing has been said about units since until one wishes a Gaussian pro- 
file, the units are arbitrary. In a PIXE-IS experiment, the abscissa would be 
sputtering charge in coulombs of inert gas ions and the ordinate would be 
X-ray photons per microcoulomb of protons. After the best integral Gaussian 
function has been chosen, conversion factors are used to convert these units 
to depth and areal density, which become depth and atomic density for the pro- 
file. The advantage in leaving the units out until this point is that the 
program can be applied to other profiling techniques. 

In line 1190 , it will be noted that five numbers are needed for the input 
statement. Line 1180 asks for R, R.S., D, T, and "1 for profile." This pro- 
vides a way of exiting from the loop and producing profile data. Normally 
when various values of range and range straggle are being tried, any other 
number except 1 is to be entered as the fifth element. When the number 1 is 
entered, the program produces the requested profile. 

To date, the program has been used to obtain profiles for annealed sam- 
ples after the range and range straggle have been determined from the 
unannealed samples. This means that the diffusion coefficient is the only 
fitting parameter. If a profile has not been obtained from an unannealed 
sample, the three fitting parameters can make the procedure quite complicated. 

The profile data are printed out based on the conversion factors and the 
values of R, R.S., D, and T. In addition to the atomic concentration (called 
simply "differential" in the program) and depth, the program provides a conver- 
sion to atomic percent (based on iron, our most commonly used base material) 
and implant dose as measured by PIXE . Note that both of these conversions are 
based on an integral zero level that was entered at statement 1320 as m. 

At this point let us return to the question of the value of infinity used 
in the integration. The numerical integration is stopped at a large enough 
value to approximate infinity such that the next incremental increase results 
in a negligible increase in the integral. This is easy since we are dealing 
with functions that decrease with d. A separate series of runs was conducted 
using a variety of values for infinity, and it was found that a value 



corresponding to 1.25 times the maximum depth (the largest value of sputtering 
charge or the point where sputtering was stopped) was more than sufficient. 
We have also chosen to divide the numerical integration into 50 steps. This 
was found to be more than enough steps to assure accuracy to within three sig- 
nificant figures. This accuracy is consistent with the experimental accuracy 
of the typical PIXE-IS experiment; thus, the program with the approximation to 
the complementary error function and the numerical integration is not the 
limiting factor. 

EXAMPLE 



What follows is an example of the use of ABC10 for determining profiles 
for iron implanted with 25 kev nickel. Figure 1 shows the data as obtained 
using the PIXE-IS technique by monitoring the Ni-L X-ray yield during 180-kev 
proton bombardment. Sputtering was performed with 1-kev argon ions. Each 

data point represents the 
average of three measure- 
ments on three separate 
samples. The solid lines 
are drawn as a result of 
using ABC10 to generate the 
integral of a Gaussian or a 
diffused Gaussian that 
agrees as close as possible 
with the data. 



.Q 

E 
o 

3 

o 
u 
o 

u 

E 

Q. 
(/> 

C 

o 
o 

Q. 

D 

_J 
UJ 

>- 
>- 
< 



3.0 




I 


25-kevNi t *- Fe 


2.5 






O As-implanted data 




tf^ 


Fa 


A Annealed (500 °C 


2.0 






40 minutes) data 






v> ^<^ 


Solid lines are fitted 


1.5 






curvesusing ABC 10 - 


1.0 








.5 




I 


I I i I 



5 10 15 20 25 

SPUTTERING CHARGE, 10 3 microcoulombs 



30 



FIGURE 1. - Integral data for nickel implanted into iron 
at 25 kev expressed as Ni-L X-ray yield 
versus sputtering charge. The solid lines 
are curves obtained using ABC 10 for fitting. 



Table 2 presents the 
computer output of the curve 
in figure 1 that represents 
the as -implanted integral 
profile and the associated 
differential concentration 
profile. It must be empha- 
sized that in this case 15 
tries were necessary before 
this curve was generated and 
the values for R and R.S. 
were obtained. After this, 
the production of the differ- 
ential profile is almost 
automatic. 



The conversion factors 
were obtained using equa- 
tions 11 and 12. Note that 
the value of 13.4 angstroms 

per 1,000 microcoulombs can be used to convert range and range straggle to 

40.2 angstroms and 53.6 angstroms, respectively. 



10 



TABLE 2. - Output of computer and input of data for evaluation of integral 
profiles for nickel implanted into iron at 25 kev and for 
obtaining profiles for as-implanted case 

ENTER VALUE OF INTEGRAL AT AND AT INFINITY? 2.62, .45 

ENTER X-RAY ABSORPTION FACTOR (ANG.-l) ? 1E-3 

ENTER STOPPING POWER (EV/ANG.), POWER FACTOR, AND EO (EV) 

? 22.5,2.067,180000 
ENTER C0NV. FACTORS: A/K AND A/C/P/MIC. ? 13.4.7.4E15 
ENTER STEP SIZE, MAXIMUM DEPTH ? 1,14 

ENTER R,R.S.,D,T,&1 FOR PROFILE ? 3,4,1,0,0 



DEPTH 


INTEGRAL 





2.62 


1 


2.36 


2 


2.12 


3 


1.85 


4 


1.58 


5 


1.32 


6 


1.09 


7 


.896 


8 


.745 


9 


.633 


10 


.557 


11 


.508 


12 


.48 


13 


.465 


14 


.459 



ENTER R,R.S.,D,T,£l FOR PROFILE ? 3,4,1,0,1 

DOSE FROM YIELD = 1.81E+16 AT0MS/CM2 

DEPTH DIFFERENTIAL ATOMIC % (FE) 

1.02E+22 12. 

13-4 1.19E+22 14. 

26.8 1.3E+22 15.4 

40.2 1.35E+22 15.9 

53.6 1.3E+22 15.4 

67- 1.19E+22 14. 

80.4 1.02E+22 12. 

93.8 8.16E+21 9.62 

107- 6.16E+21 7.26 

121. 4.37E+21 5.15 

134. 2.91E+21 3.43 

147- 1.82E+21 2.15 

161. 1.07E+21 1.26 

174. 5.91E+20 .697 

188. 3.07E+20 .362 



11 



In table 3, the values of R and R.S. obtained from the as-implanted data 
are used to obtain the curve in figure 1 that is the best match to the 
annealed data. The value of the diffusion coefficient obtained from this 
procedure is 5xl0~ 18 cm /sec. 

TABLE 3. - Output of computer and input of data for evaluation of integral 
profiles for nickel implanted into iron at 25 key and for 
obtaining profiles for annealed case 

ENTER R,R.S.,D,T,&1 FOR PROFILE 7 0,1,1,1,1 

ENTER VALUE OF INTEGRAL AT AND AT INFINITY 7 2. 2,. 25 

ENTER X-RAY ABSORPTION FACTOR (ANG.-l) 7 1E-3 

ENTER STOPPING POWER (EV/ANG.), POWER FACTOR, AND E0 (EV) 

7 22.5,2.067,180000 
ENTER C0NV. FACTORS: A/K AND A/C/P/MIC. 7 13.4.7.4E15 
ENTER STEP SIZE, MAXIMUM DEPTH 7 2,26 

ENTER R,R.S.,D,T,6l FOR PROFILE 7 3 ,4,5E- 18,2400,0 



DEPTH 


INTEGRAL 





2.2 


2 


1.99 


4 


1.82 


6 


1.65 


8 


1.48 


10 


1.32 


12 


1.17 


14 


1.02 


16 


.892 


18 


.772 


20 


.66A 


22 


.596 


24 


.487 


26 


.418 



ENTER R,R.S.,D,T,&1 FOR PROFILE 7 3,4,5E-l8,2400, 1 

DOSE FROM YIELD = 1.8E+16 AT0MS/CM2 

DEPTH DIFFERENTIAL ATOMIC % (FE) 






5.04E+21 


5.94 


26.8 


5E+21 


5.89 


53.6 


4.89E+21 


5.77 


80. 4 


4.71E+21 


5.56 


107. 


4.47E+21 


5.28 


13^. 


4.19E+21 


4.94 


161. 


3.86E+21 


4.55 


188. 


3.51E+21 


4.13 


214. 


3.14E+21 


3.7 


241. 


2.77E+21 


3.26 


268. 


2.4E+21 


2.83 


295. 


2.06E+21 


2.43 


322. 


1.74E+21 


2.05 


348. 


1.44E+21 


1.7 



12 



The quality of the agreement between the data and the curve must be 
assessed manually by calculating the deviations and averaging them to produce 
a standard deviation. If there are N total data points of value y n (x) and the 
value from the computer-generated curve is Y n (x) , then the standard deviation 
(S.D.) is 



S.D, 




(20) 



(N - 1) 



This will have the units of y n , in this case photons per microcoulomb. For 
the unannealed case S.D. = 0.062 photon/microcoulomb, and for the annealed 
case S.D. = 0.076 photon/microcoulomb. 

The subjective nature of this form of fitting curves to data is in the 
evaluation of the agreement in terms of shape. Except for the region beyond 
15X10 3 microcoulombs in the annealed case, the agreement is good. This region 
of the annealed curve is probably in such poor agreement because of nonuniform 
diffusion such as grain-boundary-enhanced diffusion and distortions in sputter 
profiles when large sputtering charges are accumulated. 

The profiles thus obtained are plotted in figure 2 as atomic percent 
nickel in iron versus depth. Note from tables 2 and 3 that the conversions 
are done by the computer and the correction factors are automatically applied. 



18 



T 



As-implanted 




25-kevNi + -*-Fe 
profiles obtained 
using ABC 10 



Annealed (500 °C, 40 minutes) 



50 



100 



150 



200 



250 



300 



DEPTH, angstroms 

FIGURE 2. - Profiles obtained for nickel implanted into iron at 25 kev as obtained using ABC10. 



13 



As a check on the use of ABC10 , it has been utilized to profile a sample 
whose profile has been well studied. A sample of arsenic-implanted silicon 
has been profiled, and the values of range and range straggle obtained agree 
reasonably well with those obtained by other experimenters (10 , 13 ) and those 
predicted by LSS theory, considering the approximate nature of LSS theory when 
applied to polycrystalline materials and considering the experimental errors 
involved. These values are shown in table 4. 

TABLE 4. - Values of range and range straggle for 30-kev As 
implanted into silicon as determined from data 
of Ludvik (13) , LSS theory (12) , Iwaki (10) , 
and the PIXE-IS technique using ABC10 





Range , 


A 


Range straggle, 


A 


PIXE-IS 


260 




180 




LSS 


212 




83 






210 




210 






210 




75 





CONCLUSIONS 



It has been shown that ABC10 can be a useful tool for interpreting inte- 
gral Gaussian profiles, particularly for ion-implanted metals. The program 
utilizes manual fitting of integral profiles with range, range straggle, and 
diffusion coefficient as parameters resulting in Gaussian or diffused Gaussian 
(Fick's law) profiles. The program takes account of X-ray absorption and pro- 
ton energy loss and includes conversion factors that can result in a quantita- 
tive profile. 



14 

REFERENCES 

1. Abramowitz, M. , and I. A. Stegun. Handbook of Mathematical Functions. 

Dover Press, New York, 1965, p. 299. 

2. Andersen, H. H. , and J. F. Ziegler. Hydrogen Stopping Powers and Ranges 

in All Elements. Pergamon Press, New York, 1977, 317 pp. 

3. Churchill, R. V. Fourier Series and Boundary Value Problems. McGraw- 

Hill Book Co., New York, 1963, p. 156. 

4. Covino, B. S., Jr., P. B. Needham, Jr., and G. R. Conner. Anodic Polar- 

ization Behavior of Fe-Ni Alloys Fabricated by Ion Implantation. 
J. Electrochem. Soc, v. 125, No. 3, March 1978, pp. 370-372. 

5. Covino, B. S., Jr., B. D. Sartwell, and P. B. Needham, Jr. Anodic Polar- 

ization Behavior of Fe-Cr Surface Alloys Formed by Ion Implantation. 
J. Electrochem. Soc, v. 125, No. 3, March 1978, pp. 366-369. 

6. Crank, J. The Mathematics of Diffusion. University Press, New York, 

1957, 259 pp. 

7. Fox, A. H. Fundamentals of Numerical Analysis. Ronald Press, New York, 

1963, p. 58. 

8. General Services Administration Computer Time-Sharing Service. Applica- 

tions Library Operating Instructions. Federal Data Processing Center, 
Atlanta, Ga . , August 1973, 358 pp. 

9. Henke , B. L. , and E. S. Ebisu. Low Energy X-Ray and Electron Absorption 

Within Solids (100-1500 eV Region). Ch. in Advances in X-Ray Analysis, 
v. 17, ed. by C. L. Grant, C. S. Barrett, J. B. Newkirk, and C. 0. Rund. 
Plenum Press, New York, 1974, pp. 150-213. 

10. Iwaki , M. , K. Gamo, K. Masuda , S. Namba , S. Ishihara , and I. Kimurd. 

Concentration Profiles of Arsenic Implanted in Silicon. Ch. in Ion 
Implantation in Semiconductors and Other Materials, ed. by B. L. Crowder, 
Plenum Press, New York, 1973, pp. 111-118. 

11. Khan, J. M. , D. L. Potter, and R. D. Worley. Proposed Method for Micro- 

gram Surface Density Measurements by Observation of Proton-Produced 
X-Rays. J. Appl. Phys . , v. 37, No. 2, February 1966, pp. 564-567. 

12. Lindhard , J., J. M. Scharf f , and H. E. Schiott. Atomic Collisions, II. 

Range Concept and Heavy Ion Ranges. Kgl. Danske Vid. Selsk. , Matt-Fys. 
Medd., v. 33, No. 14, 1963, pp. 42-52. 

13. Ludvik, S., L. Scharpen, and H. E. Weaver. Measurement of Arsenic Implan- 

tation Profiles in Silicon Using an Electron Spectroscopic Technique. 
Ch. in Ion Implantation in Semiconductors, ed. by S. Namba. Plenum 
Press, New York, 1975, pp. 155-162. 



15 



14. Mayer, J. W. , and E. Rimini. Ion Beam Handbook for Material Analysis. 

Academic Press, New York, 1977, p. 316. 

15. Oechsner, H. Sputtering - A Review of Some Recent Experimental and Theo- 

retical Aspects. Appl. Phys . , v. 8, 1975, pp. 185-191. 

16. Winterbon, K. B. Ion Implantation Range and Energy Deposition Distribu- 

tions. Plenum Press, New York, v. 2, 1975, 341 pp. 



16 



APPENDIX.— LISTING OF ABCIO 

The following listing of ABCIO was obtained directly from storage in the 
GSA RAMUS system. Some of the symbols are defined in table 1. 

100 REM ABC 10 

110 PRINT 

120 PRINT "IF R=0, RESET ALL. IF R.S.=0, RESET STEP AND MAX. DEPTH." 

130 PRINT 

HO DIM U (10),R(10) 

150 LET 1=0 

160 FOR 1=1 TO 5 

170 READ U(l),R(l) 

180 NEXT I 

190 GO TO 1310 

200 PRINT "ENTER STEP SIZE, MAXIMUM DEPTH"; 

210 INPUT T6,V 

220 LET L=0 

225 FOR L=0 TO V STEP T6 

230 LET A0=L 

240 LET B0=1.25-V 

250 LET N0=50 

260 IF L=0 THEN 1170 

270 LET X=X 

320 LET Al=. 34802 

330 LET A2=-. 09588 

340 LET A3=. 74786 

350 IF T=0 THEN 720 

360 DEF FNE(Y,Z)=(Al*Y+A2*Y!2+A3*Y+3)"EXP(-l*Zt2) 

370 LET B2=1+((2*D*T)/((C+2)*1E-16)) 

380 LET B1=1/(2*(C+2)*B2) 

390 LET B3=(lE-8)/(2*SQR(D*T)*S0_R(B2)) 

400 LET B4=((D*T*B)*lE8)/((Ct2)*SQ.R(D*T)*S0_R(B2)) 

MO LET B5=A/(2-'SQR(B2)) 

420 LET X4=-1*B3*X-B4 

430 LET X5=B3*X-B4 

440 LET U4=EXP(-l*Bl*(X-B)+2) 

450 LET U5 =EXP(-l*Bl*(X+B)+2) 

460 IF X4<0 THEN 560 

470 IF X5<0 THEN 620 

480 IF X4>0 THEN 680 

490 IF X5<0 THEN 530 

500 LET T2=1/(1+.47047*X5) 

510 LET U7 =FNE(T2,X5) 

520 LET U2=U5-U7 

530 LET Y=G*B5*(U1+U2) 

535 IF E=l THEN 1430 

550 GO TO 790 

560 LET X4=-1*X4 

570 LET Tl=l/(l+.47047*X4) 

580 LET U6 =2-FNE(T1,X4) 

590 LET Ul =U4*U6 

600 LET X4=-l -'-X4 

610 GO TO 470 

620 LET X6=-1*X5 

630 LET T2=l/(l + .47047- ; X5) 

640 LET U7 =2-FNE(T2,X5) 

650 LET U2=U5"--U7 

660 LET X5=-l-X5 

670 GO to 430 

680 LET Tl = l/(l + .47047 ; -X4) 

690 LET U6 =FNE(T1,X4) 

700 LET U1=U4*U6 



17 



710 IP X4>0 THEN 530 

720 LET B2=1+((2*D*T)/((C+2)*1E-16)) 

730 LET B1=1/(2*C+2*B2) 

740 LET B5=A/2 

750 LET Ul=2*EXP((-l*Bl)*(X-B)+2) 

760 LET U2=0 

770 GO TO 530 

790 IF J=l THEN 980 

800 IF J=2 THEN 1020 

810 LET A8=A0 

820 LET ZO = 

830 LET D8=(B0-A0)/N0 

840 LET B8=A8 

850 LET B8=B8+D8 

860 IF B8<=B0 THEN 890 

870 IF ZOO THEN 1070 

880 LET B8=B0 

890 LET C1 = .5- : (b3+A8) 

900 LET C2=B8-A8 

910 LET S=0 

920 LET 1=0 

930 LET 1=1+1 

940 LET W=C2*J(l) 

950 LET X=C1+W 

960 LET J=l 

970 GO TO 270 

980 LET S=S+R(I)*Y 

990 LET X=C1-W 

1000 LET J=2 

1010 GO TO 270 

1020 LET S=S+R(I)*Y 

1030 IF l<5 THEN 930 

1040 LET ZO =Z0 + S*C2 

1050 LET A8=B8 

1060 GOTO 850 

1070 IF L=0 THEN 1270 

1080 LET 0_=ZO+M 

1085 LET Q6=(V-L)/2 

1086 LET Q7=EXP(-l*0_6*0_8*K) 

1087 LET Gii=((G3=(Gl*K*Q6))/G3)+G2 
1090 PRINT L,Q*0_7*G4 

1100 LET J=0 

1110 IF Z0<(P/1E4) THEN 220 

1120 IF (V-L)<T6 THEN 1 460 

1130 NEXT L 

1140 DATA 744371695E-10, 1477621 12E-9,2l6657697E-9,134633360E-9 

1150 DATA 339704784E-9,109543181E-9,432531683E-9,747256746E-10 

1160 DATA 486953264E-9,333356722E-!0 

1170 PRINT 

ll80 PRINT "ENTER R,R .S . ,D ,T,£l FOR PROFILE"; 

1190 INPUT B,C,D,T,E 

1200 IF B=0 THEN 1310 

1210 IF C=0 THEN 200 

1220 IF E=l THEN 13"0 

1230 PRINT 

1240 PRINT "DEPTH" ."INTEGRAL" 



1250 LET G=l 

1255 LET A=l 

1260 GO TO 270 

1270 LET H6=EXP(-1*Q8*K*(V/2)) 

1272 LET H7=((G3-(Gl*K*(V/2)))/G3)*G2 

1273 IF E=l THEN 1384 

1275 LET G-(P-M)/(Z0*H6*H7) 

1280 SET DIGITS 3 

1290 PRINT L,P 

1300 GO TO 1100 

1310 PRINT "ENTER VALUE OF INTEGRAL AT AND AT INFINITY"; 

1320 INPUT P,M 

1325 IF M> P THEN 1310 

1345 PRINT "ENTER X-RAY ABSORPTION FACTOR (ANG.-l)"; 

1 346 INPUT 0.8 

13^7 PRINT "ENTER STOPPING POWER (EV/ANG.), POWER FACTOR, AND EO (EV)" 

1348 INPUT G1.G2.G3 

1360 PRINT "ENTER CONV. FACTORS: A/K AND A/C/P/MIC"; 

1370 INPUT K,H 

1375 GO TO 200 

1380 SET DIGITS 3 

1381 PRINT 

1382 GO TO 1270 

1384 PRINT "DOSE FROM YIELD =";H* (P-M)/H6*H7) ;"AT0MS/CM2" 

1385 PRINT 

1386 PRINT "DEPTH", "DIFFERENTIAL", "ATOMIC S(FE)" 

1387 PRINT 

1388 LET C6=(2-FNE (0/(1+ (. 47047)* (B/(l.4l4*C)))),(B/(l.4l4*C)))) 

1389 LET A«H*(P-M)/(H6*H7*2.5066E-8*C*K*C6) 

1390 LET G=l 

1400 FOR X=0 TO V STEP T6 

1415 GO TO 320 

1430 PRINT X*K,Y,100*(Y/8.48E22) 

1440 IF X=V GO TO 1170 

1450 NEXT X 

1460 LET L=0 

1470 GO TO 220 

1480 END 



^> 



X % 



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